(f) $$ f: \mathbb{R}^{2} \rightarrow \mathbb{R}:(x, y) \mapsto x^{2}+y^{2} $$

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Certainly! Let's analyze the function you've provided:

$$
f: \mathbb{R}^{2} \rightarrow \mathbb{R} \quad \text{defined by} \quad f(x, y) = x^{2} + y^{2}
$$

### 1. **Domain and Codomain**
- **Domain**: The function $$ f $$ is defined for all pairs $$(x, y)$$ in $$\mathbb{R}^{2}$$. This means any real number values for $$x$$ and $$y$$ are permissible.
- **Codomain**: The output of the function is a real number $$ \mathbb{R} $$.

### 2. **Graphical Representation**
- The graph of $$ f(x, y) = x^{2} + y^{2} $$ is a **paraboloid** in three-dimensional space. Specifically, it's a **circular paraboloid** because the cross-sections parallel to the $$x$$-$$y$$ plane are circles.

### 3. **Level Sets**
- **Level sets** (or **contour lines**) for this function are given by:
  $$
  x^{2} + y^{2} = c
  $$
  where $$c$$ is a constant.
- For $$c > 0$$, these are circles centered at the origin with radius $$\sqrt{c}$$.
- For $$c = 0$$, it's just the point $$(0,0)$$.
- There are no real points for $$c < 0$$ since the sum of squares is always non-negative.

### 4. **Differentiability**
- $$ f(x, y) = x^{2} + y^{2} $$ is **infinitely differentiable** (i.e., smooth) over its entire domain $$\mathbb{R}^{2}$$.

### 5. **First Partial Derivatives**
- The first partial derivatives of $$ f $$ are:
  $$
  \frac{\partial f}{\partial x} = 2x, \quad \frac{\partial f}{\partial y} = 2y
  $$
- These derivatives exist everywhere in $$\mathbb{R}^{2}$$, reinforcing the function's differentiability.

### 6. **Critical Points and Extrema**
- **Critical Points**: Points where both first partial derivatives are zero.
  $$
  2x = 0 \quad \text{and} \quad 2y = 0 \implies (0, 0)
  $$
  So, the only critical point is at $$(0, 0)$$.

- **Nature of Critical Point**:
  - To determine whether $$(0, 0)$$ is a minimum, maximum, or saddle point, we can examine the second partial derivatives.

### 7. **Second Partial Derivatives and the Hessian Matrix**
- Second partial derivatives:
  $$
  \frac{\partial^{2} f}{\partial x^{2}} = 2, \quad \frac{\partial^{2} f}{\partial y^{2}} = 2, \quad \frac{\partial^{2} f}{\partial x \partial y} = 0
  $$
  
- The **Hessian matrix** $$ H $$ is:
  $$
  H = \begin{bmatrix}
  2 & 0 \
  0 & 2
  \end{bmatrix}
  $$
  
- **Determinant of Hessian**:
  $$
  \det(H) = (2)(2) - (0)(0) = 4 > 0
  $$
  
- Since the determinant is positive and the second partial derivatives are positive, the critical point at $$(0, 0)$$ is a **local minimum**.

### 8. **Global Behavior**
- The function $$ f(x, y) = x^{2} + y^{2} $$ is **convex** over $$\mathbb{R}^{2}$$, meaning any local minimum is also a global minimum.
- As $$ \sqrt{x^{2} + y^{2}} $$ (the Euclidean distance from the origin) increases, $$ f(x, y) $$ tends to infinity. Therefore, the function has no upper bound and attains its minimum value of $$0$$ at the origin.

### 9. **Applications**
- **Optimization**: Such quadratic functions are fundamental in optimization problems, especially in least squares fitting.
- **Physics**: Represents potential energy in systems with quadratic potential wells.
- **Geometry**: Defines circles and spheres in higher dimensions.

### 10. **Polar Coordinates Representation**
- By switching to polar coordinates $$(r, \theta)$$, where $$ x = r \cos\theta $$ and $$ y = r \sin\theta $$, the function simplifies to:
  $$
  f(r, \theta) = r^{2}(\cos^{2}\theta + \sin^{2}\theta) = r^{2}
  $$
- This shows that $$ f $$ depends only on the distance from the origin, not on the angle $$ \theta $$.

### 11. **Gradient and Directional Derivatives**
- **Gradient Vector**:
  $$
  \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (2x, 2y)
  $$
  This vector points radially outward from the origin and its magnitude increases with distance from the origin.

- **Directional Derivative**:
  - The directional derivative of $$ f $$ in the direction of a unit vector $$ \mathbf{u} = (u_1, u_2) $$ is:
    $$
    D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u} = 2x u_1 + 2y u_2
    $$
  - This measures the rate at which $$ f $$ changes as you move in the direction of $$ \mathbf{u} $$.

### 12. **Laplace's Equation**
- The function $$ f $$ satisfies Laplace's equation:
  $$
  \Delta f = \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} = 2 + 2 = 4 \neq 0
  $$
  Hence, it is **not** a harmonic function.

### 13. **Symmetry**
- $$ f $$ is **rotationally symmetric** about the origin. Rotating the input coordinates results in the same output value, reflecting the function's dependence solely on the distance from the origin.

### 14. **Contours and Equipotential Lines**
- As mentioned earlier, the contours $$ x^{2} + y^{2} = c $$ are circles. In physics, if $$ f $$ represents a potential field, these circles would be **equipotential lines** where the potential is constant.

### 15. **Integrability**
- The function is **integrable** over any compact subset of $$\mathbb{R}^{2}$$. Its integrals can be easily evaluated using polar coordinates.

### 16. **Fourier Transform**
- While not always necessary for basic analysis, the Fourier transform of quadratic functions involves Gaussian integrals, which are fundamental in various fields like signal processing and quantum mechanics.

### 17. **Eigenvalues and Eigenvectors**
- If we consider the quadratic form associated with $$ f $$, represented by the matrix:
  $$
  A = \begin{bmatrix}
  1 & 0 \
  0 & 1
  \end{bmatrix}
  $$
  - The eigenvalues are both $$1$$, and the eigenvectors can be any orthonormal basis for $$\mathbb{R}^{2}$$, reflecting the function's isotropy.

### 18. **Optimization Problem Example**
- **Minimization**: Find the minimum of $$ f(x, y) $$ subject to certain constraints. For instance, minimizing distance squared from the origin leads directly to the critical point identified earlier.

### Summary

The function $$ f(x, y) = x^{2} + y^{2} $$ is a fundamental quadratic form with numerous applications in mathematics, physics, and engineering. Its simplicity allows for easy analysis of its properties, making it a staple example in various theoretical and applied contexts.
The function $$ f(x, y) = x^{2} + y^{2} $$ is a quadratic form that maps any point in the plane to its squared distance from the origin. It is smooth, has a minimum at the origin, and its contours are circles centered at the origin.

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